SingleLenses.md

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Single lenses

The lens equation for a single lens is

$$ u=x-\frac{1}{x}$$

For a given lens-source angular separation $u$ there are two images $x$ solving this equation. We note that all angular distances are in units of the Einstein angle $\theta_E$.

Point-Source-Point-Lens

If the source is point-like, the magnification is given by the famous Paczynski formula

$$ \mu = \frac{u^2+2}{u\sqrt{u^2+4}}$$

In VBMicrolensing, this formula is obtained through the function PSPLMag as follows:

VBMicrolensing VBM;
double Mag,u;

u=0.1;  // Source-lens separation in Einstein radii
Mag = VBM.PSPLMag(u);
printf("PSPL Magnification = %lf", Mag); // Output should be 10.037...

Extended-Source-Point-Lens

For extended sources, the magnification depends on $\rho$, the source radius normalized to the Einstein angle, and can be calculated through elliptic integrals. In order to make VBMicrolensing as fast as possible, we provide pre-calculated tables in the file "ESPL.tbl". This file should be loaded before any calculations involving Extended-Source-Point-Lenses (ESPL).

VBMicrolensing VBM;
double Mag, u, rho;

VBM.LoadESPLTable("ESPL.tbl"); // Load the pre-calculated table (you only have to do this once)
// The ESPL.dat file is located inside the data folder; copy it to your directory.

u = 0.1; // Source-lens separation in Einstein radii
rho = 0.01; // Source radius in units of the Einstein angle

Mag = VBM.ESPLMag2(u, rho); // Call to the ESPLMag2 function with these parameters
printf("\nMagnification of Extended-source-point-lens = %lf\n", Mag);  // Output should be 10.050.....

The current range for the pre-calculated table is $10^{-4} \leq \rho \leq 10^{+2}$. Sources smaller than the minimum are considered equal to the minimum. Sources larger than the maximum generate an error message.

By default, VBMicrolensing works with uniform sources. We will introduce Limb Darkening in a later section: arbitrary Limb Darkening laws can be implemented in VBMicrolensing.

Astrometry

For a Point-Source, in the reference frame in which the lens is in the origin, the centroid of the images is simply

$$ \bar x = \frac{u}{u^2+2} + u$$

If you need astrometry calculations together with magnification, you have to turn astrometry on by VBM.astrometry = true and read the results in VBM.astrox1. This works in the same way for PSPLMag and ESPLMag2.

VBMicrolensing VBM;
double Mag,u,rho;

VBM.LoadESPLTable("ESPL.tbl"); // Load the pre-calculated table (you only have to do this once)

u = 0.1; // Source-lens separation in Einstein radii
rho = 0.01; // Source radius in units of the Einstein angle

VBM.astrometry = true; // We want astrometry

Mag = VBM.ESPLMag2(u, rho); // Call to the ESPLMag2 function with these parameters
printf("\nMagnification of Extended-source-point-lens = %lf\n", Mag);  // Output should be 10.050.....
printf("\nCentroid shift = %lf\n", VBM.astrox1 - u);  // Output should be 0.0493.....

Note that VBM.astrox1 reports the centroid position with respect to the lens. The centroid position with respect to the source is VBM.astrox1 - u.

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